In the following we point out how considering higher Picard iterates allows to construct solutions for very slowly decaying initial data. The arguments are similar to [1]} and [2]}, albeit with the difference that the Duhamel integral has a stronger smoothing effect. We consider the cubic nonlinear wave equation in \(d\)  dimensions:
\(\left\lbrace \!\begin{array}{cl}\partial _t^2 u - \Delta _{x} u &= \pm |u|^2 u, \quad (t,x) \in \mathbb {R}\times \mathbb {R}^d, \; d \ge 2,\\u(0) &= f_1 \in X, \quad \dot{u}(0) = f_2 \in Y,\end{array} \right.\) 
